Transforming Polynomial Functions

Transcription

1 5-9 Transforming Polnomial Functions Content Standards F.BF.3 Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative) find the value of k given the graphs. Also F.IF.7.c, F.IF., F.IF.9 Objective To appl transformations to graphs of polnomials Remember ou transformed the graphs of quadratic and absolute value functions. The graph of the parent cubic function f () 5 3 is one of the graphs at the right. The other graph is a transformation g of the parent function. What is an equation for g? How do ou know? O Lesson Vocabular power function constant of proportionalit MATHEMATICAL PRACTICES Recall that ou can obtain the graph of an quadratic function from the graph of the parent quadratic function, 5, using one or more basic transformations. You will find that this is not true of cubic functions. Essential Understanding The graph of the function 5 af ( h) k is a vertical stretch or compression b the factor u a u, a horizontal shift of h units, and a vertical shift of k units of the graph of 5 f (). Problem Transforming 5 3 What is an equation of the graph of 5 3 under a vertical compression b the factor followed b a reflection across the -ais, a horizontal translation 3 units to the right, and then a vertical translation units up? How is translating this cubic function like translating a quadratic function? In each case ou replace with h to translate h units to the right. Step Step Step 3 Step Multipl b to compress Multipl b to reflect Replace with 3 to translate horizontall ( 3) 3 Add to translate verticall. 5 ( 3) 3 5 ( 3) 3 Lesson 5-9 Transforming Polnomial Functions 339

2 Got It?. What is an equation of the graph of 5 3 under a vertical stretch b the factor followed b a horizontal translation 3 units to the left and then a vertical translation units down? The graph shows 5 3 and the graphs that result from the transformations in Problem. In general, 5 a( k represents all of the cubic functions ou can obtain b stretching, compressing, reflecting, or translating the cubic parent function h)3 ( 3)3 O Problem Finding Zeros of a Transformed Cubic Function Multiple Choice If a, h, and k are real numbers and a u 0, how man distinct real zeros does 5 a( h)3 k have? How can ou find the zeros of a function? Set the function equal to 0 and solve for. 0 aa hb 3 k 5 0 aa hb 3 5 k is a zero means it is an -intercept, so 5 0. Subtract k from each side. k Divide each side b a. 3k Take the cube root of each side. 3 k Solve for. A hb 3 5 a h 5 Åa 5 Åa h 3 Disregarding multiplicities, the function has a single real zero. The correct answer is B. Got It?. What are all the real zeros of the function 5 3( )3 6? Problems and together illustrate that the graph of an offspring function of the parent cubic function 5 3 has onl one -intercept. The graph of the cubic function has three -intercepts. You cannot obtain this function or others like it b transforming the parent cubic function 5 3 using stretches, reflections, and translations. Similarl, some quartic functions are simple transformations of 5 and some are not. 30 (, 0) (3, 0) (, 0) 6 Chapter 5 Polnomials and Polnomial Functions smase_cc_0509.indd 30 3/9/ :

3 Problem 3 Constructing a Quartic Function with Two Real Zeros What is a quartic function with onl two real zeros, 5 5 and 5 9? Method Use transformations. First, find a quartic with zeros at. Translate the basic quartic 6 units down: 5 S is 7 units to the right of. Translate 7 units to the right. 5 6 S 5 ( 7) 6 A quartic function with its onl real zeros at 5 and 9 is 5 ( 7) 6. Method Use algebraic methods. 6 ( 7) 6 What should ou use for Q()? Choose Q () to be a quadratic with no real zeros. Keep it simple, such as. 5 ( 5)( 9)? Q () 5 ( 5)( 9)A B 5 A 5BA B Another quartic function with its onl real zeros at 5 and 9 is Make Q() a quadratic with no real zeros. Got It? 3. a. What is a quartic function f () with onl two real zeros, 5 0 and 5 6? b. Reasoning Does the quartic function f () have the same zeros? Eplain. The offspring of the parent function 5 is a subfamil of all quartic polnomials. This subfamil consists of quartics of the form 5 a( h) k. These functions also belong to another categor of polnomials, and in this categor ou can generate families as usual. Ke Concept Power Functions Definition A power function is a function of the form 5 a? b, where a and b are nonzero real numbers. Eamples If the eponent b in 5 a b is a positive integer, the function is also a monomial function. If 5 a b describes as a power function of, then varies directl with, or is proportional to, the bth power of. The constant a is the constant of proportionalit. Power functions arise in man real-world contets related to the concept of direct variation, which ou studied in Chapter. Lesson 5-9 Transforming Polnomial Functions 3 mase_cc_0509.indd 3 3/9/ :

4 Problem Modeling With a Power Function STEM How is this problem like the direct variation ou studied in Chapter? With the direct variation, 5 k, ou use a first power. Here ou use a third power. Wind-Generated Power Wind farms are a source of renewable energ found around the world. The power P (in kilowatts) generated b a wind turbine varies directl as the cube of the wind speed v (in meters per second). The picture shows the power output of one turbine at one wind speed. To the nearest kilowatt, how much power does this turbine generate in a 0 m/s wind? Wind m/s Electric Power 600 kw The formula for P as a power function of v is P 5 a? v 3. From the picture, P when v 5, or a? 3. Solve for a a? 3 Use values of P and v to find a a a <.79 P <.79v 3. P <.79? Use the value of a in the original formula. Substitute 0 for v and simplif. This turbine generates about 7 kw of power in a 0 m/s wind. Got It?. Another turbine generates 0 kw of power in a mi/h wind. How much power does this turbine generate in a 0 mi/h wind? Lesson Check Do ou know HOW? Find all the real zeros of each function.. 5( 3) 3. 5( 5) ( )3 3 Do ou UNDERSTAND? MATHEMATICAL PRACTICES. Vocabular Is the function an eample of a power function? Eplain. 5. Error Analsis Your friend sas that he has found a wa to transform the graph of 5 3 to obtain three real roots. Using the graph of the function, eplain wh this is impossible. 6. Compare and Contrast How are the graphs of 5 3 and 5 3 alike? How are the different? What transformation was used to get the second equation? 3 Chapter 5 Polnomials and Polnomial Functions mase_cc_0509.indd 3 6// 0

5 Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Determine the cubic function that is obtained from the parent function 5 3 after each sequence of transformations. See Problem. 7. a vertical stretch b a factor of 3;. a vertical stretch b a factor of ; a reflection across the -ais; a vertical translation units up; a vertical translation units up; and a horizontal translation 3 units left and a horizontal translation unit right 9. a reflection across the -ais; 0. a vertical translation 3 units down; a vertical translation unit down; and a horizontal translation units right and a horizontal translation 5 units left. a vertical stretch b a factor of 3;. a vertical stretch b a factor of 5 3 ; a reflection across the -ais; a reflection across the -ais; a vertical translation 3 unit up; a vertical translation units down; and a horizontal translation unit left and a horizontal translation 3 units right Find all the real zeros of each function ( ) 3. 5 ( 7)3 See Problem Q 5 R ( 3) ( ) ( 5) 3 0 Find a quartic function with the given -values as its onl real zeros and and 5. 5 and and and and 5 5. Cooking Th e number of pepperoni slices that Kim puts on a pizza varies directl as the square of the diameter of the pizza. If she puts 5 slices on a 0 0 diameter pizza, how man slices should she put on a 6" diameter pizza? See Problem 3. See Problem. B Appl 6. Volume The amount of water that a spherical tank can hold varies directl as the cube of its radius. If a tank with radius 7.5 ft holds 767 ft 3 of water, how much water can a tank with radius 6 ft hold? 7. Think About a Plan Th e kinetic energ generated b a 5 lb ball is represented b the formula K 5 (5)v. If the ball is thrown with a velocit of 6 ft/sec, how much kinetic energ is generated? What does 5 represent in the function? What number should ou substitute for v? Lesson 5-9 Transforming Polnomial Functions 33 mase_cc_0509.indd 33 3/9/ :

6 Determine whether each function can be obtained from the parent function, 5 n, using basic transformations. If so, describe the sequence of transformations ( 3) Determine the transformations that were used to change the graph of the parent function 5 3 to each of the following graphs (0, 5) (, ) (0, 3) (, ) O O (, ) ( 3, ) (, 6) O (3, ) (, 3) (, 6) (, ) (, ) O O (, ) (0, ) (0, 5) (, 0) (, ) O (6, 6) STEM 3. Compare the function to the function shown in the graph at the right. Which function has a greater vertical stretch factor? Eplain. 39. Phsics Th e formula K 5 mv represents the kinetic energ of an object. If the kinetic energ of a ball is 0 lb-ft /s when it is thrown with a velocit of ft/s, how much kinetic energ is generated if the ball is thrown with a velocit of ft/s? 0. Reasoning Eplain wh the basic transformations of the parent function 5 5 will onl generate functions that can be written in the form 5 a( h) 5 k.. Reasoning Eplain wh some quartic polnomials cannot be written in the form 5 a( h) k. Give two eamples. (, ) (0, 0) (, ) 3 Chapter 5 Polnomials and Polnomial Functions mase_cc_0509.indd 3 3/9/ :

7 C Challenge STEM. Reasoning Find a sequence of basic transformations b which the polnomial function can be derived from the cubic function Phsics For a constant resistance R (in ohms), the power P (in watts) dissipated across two terminals of a batter varies directl as the square of the current I (in amps). If a batter connected in a circuit dissipates watts of power for amps of current flow, how much power would be dissipated when the current flow is 5 amps?. Writing Give an argument that shows that ever polnomial famil of degree n. contains polnomials that cannot be generated from the basic function 5 n b using stretches, compressions, reflections, and translations. SAT/ACT Short Response Standardized Test Prep Use the graph to answer questions Which equation does the graph represent? 5 ( ) 5 ( ) 5 ( ) 5 ( ) 6. If 5 f () is an equation for the graph, what are factors of f ()? ( ) and ( 3) ( ) and ( 3) ( ) and ( 3) ( ) and ( 3) 7. If 5 a b c is an equation for the graph, what tpe of number is its discriminant? O Mied Review Find a polnomial function whose graph passes through the given points.. (, ), (0, ), (, ), (, ) 9. (, 7), (0, 3), (, 5), (3, 63) See Lesson 5-. Write an equation of each line. 50. slope 5 5 ; through (, ) 5. slope 53; through (, ) See Lesson -. Determine whether each relation is a function. 5. 5(0, ), (, 3), (, 3), (3, 3)6 See Lesson (, 0), (7, 0), (, ), (7, )6 Get Read! To prepare for Lesson 6-, do Eercises Factor each epression See Lesson Lesson 5-9 Transforming Polnomial Functions 35 mase_cc_0509.indd 35 3/9/ :